Strong 1-Boundedness of Unimodular Free Orthogonal Quantum Groups

Abstract

Recently, Brannan and Vergnioux showed that the free orthogonal quantum group factors $\mathcal{L}\mathbb{F}{O}_M$ have Jung's strong 1-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in 2N dimensions $J_{2N}$. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in 1-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a 1-bounded set without losing 1-boundedness. In particular this allows us to include the character of the fundamental representation, proving strong 1-boundedness.

arXiv comment15 pages, comments welcome

References in corpus (10)

• Quantum Symmetry Groups of Finite Spaces
• Integration over compact quantum groups
• CCAP for universal discrete quantum groups
• Orientation of quantum Cayley trees and applications
• Hochschild homology of Hopf algebras and free Yetter-Drinfeld resolutions of the counit
• The Connes embedding property for quantum group von Neumann algebras
• Orthogonal free quantum group factors are strongly 1-bounded
• L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups
• Strong asymptotic freeness for free orthogonal quantum groups
• The Rank Theorem and $L^2$-invariants in Free Entropy: Global Upper Bounds